U.S. patent number 5,164,647 [Application Number 07/632,842] was granted by the patent office on 1992-11-17 for multivariable adaptive vibration canceller.
This patent grant is currently assigned to Westinghouse Electric Corp.. Invention is credited to Clinton W. Moulds, III.
United States Patent |
5,164,647 |
Moulds, III |
November 17, 1992 |
Multivariable adaptive vibration canceller
Abstract
A system for minimizing periodically induced vibration in a
mechanical structure, particularly rotary shaft apparatus that is
not limited to the use of any particular number or type of forcers
or sensors and requires no knowledge of the mechanical structure's
transfer function. In accordance with a preferred embodiment the
system senses periodically induced vibration utilizing a plurality
of sensors and produces a complex output representative of the
algebraic sum of the vibrations sensed for each harmonic and
counteracting the vibration with a plurality of actuators
operatively coupled to the structure by producing a counteracting
vibration therein in response to a complex input signal for each
actuator in the form of a matrix of system responses to vibrational
inputs at selected harmonics of interest. More specifically, the
complex inputs for the actuators are produced on the basis of an
adaptation cycle wherein the effect of an adjustment of the complex
inputs on the complex outputs is utilized to determine the nature
of a successive adjustment of the complex inputs utilizing a
control algorithm.
Inventors: |
Moulds, III; Clinton W.
(Millersville, MD) |
Assignee: |
Westinghouse Electric Corp.
(Pittsburgh, PA)
|
Family
ID: |
24537178 |
Appl.
No.: |
07/632,842 |
Filed: |
December 24, 1990 |
Current U.S.
Class: |
318/561 |
Current CPC
Class: |
G05D
19/02 (20130101) |
Current International
Class: |
G05D
19/00 (20060101); G05D 19/02 (20060101); G05B
013/00 () |
Field of
Search: |
;318/561,640,128
;364/158 ;384/448 ;248/550 ;267/136 |
References Cited
[Referenced By]
U.S. Patent Documents
Primary Examiner: Shoop, Jr.; William M.
Attorney, Agent or Firm: Williamson; J. K.
Claims
I claim:
1. A multivariable adaptive vibration canceller for cancelling
periodically induced vibration in a mechanical structure
comprising:
a plurality of actuators operatively coupled to the structure for
producing a counteracting vibration therein in response to a
complex input signal for each actuator, the structure having a
dynamic structural system characteristic in the form of a matrix of
system responses to vibrational inputs at selected harmonics of
interest;
a plurality of sensors, at least equal to the number of actuators
coupled to the structure for sensing periodically induced vibration
and actuator induced vibration, said sensors producing a complex
output representative of the algebraic sum of the vibrations sensed
thereby for each harmonic;
multivariable adaptive vibration cancellation processor means
responsive to complex outputs of the sensors for producing the
complex inputs to the actuators, said vibration cancellation
processor means producing said inputs on the basis of an adaptation
cycle wherein the effect of an adjustment of said complex inputs on
said complex outputs is utilized to determine the nature of a
successive adjustment of the complex inputs independent of
knowledge of said dynamic structural system characteristic.
2. A multivariable adaptive vibration canceller according to claim
1, wherein said cancellation processor means includes means for
performing a control algorithm to determine the complex input, said
control algorithm being:
where x.sub.j is a nominal value of the complex input during a
current adaptation cycle j, x.sub.j+1 is the nominal value of the
complex input vector for the next adaptation cycle j+1, E.sup.2 ( )
is the square of the complex output measured with a perturbed
(x.sub.j +u.sub.j) and an unperturbed (x.sub.j) complex input,
u.sub.j is the jth sample of a random vector with covariance
.sigma..sup.2 I, and .beta. is a positive constant used to control
the speed of adjustment of x.
3. A multivariable adaptive vibration canceller according to claim
1, wherein the vibration canceller processor means changes the
value of x.sub.j+1 in the direction of u.sub.j when a decrease in
E.sup.2 occurs in response to adding of u.sub.j to the complex
input x.sub.j, and changes x.sub.j+1 in a direction opposite to
u.sub.j when E.sup.2 increases in response to adding of u.sub.j to
the complex input x.sub.j.
4. A multivariable adaptive vibration canceller according to claim
3, wherein said structure includes a rotating shaft.
5. A multivariable adaptive vibration canceller according to claim
2, wherein said structure includes a rotating shaft.
6. A multivariable adaptive vibration canceller according to claim
1, wherein said structure includes a rotating shaft.
7. A multivariable adaptive vibration canceller according to claim
1, wherein said sensors comprise accelerometers.
8. A multivariable adaptive vibration canceller according to claim
2, wherein said sensors comprise accelerometers.
9. A multivariable adaptive vibration canceller according to claim
3, wherein said sensors comprise accelerometers.
10. A multivariable adaptive vibration canceller according to claim
4, wherein said sensors comprise accelerometers.
11. A multivariable adaptive vibration canceller according to claim
6, wherein said sensors comprise accelerometers.
12. A method for adaptively cancelling periodically induced
vibration in a mechanical structure comprising the steps of:
sensing periodically induced vibration utilizing a plurality of
sensors and producing a complex output representative of the
algebraic sum of the vibrations sensed thereby for each
harmonic;
counteracting said vibration with a plurality of actuators
operatively coupled to the structure by producing a counteracting
vibration therein in response to a complex input signal for each
actuator in the form of a matrix of system responses to vibrational
inputs at selected harmonics of interest;
producing the complex inputs for the actuators on the basis of an
adaptation cycle wherein the effect of an adjustment of said
complex inputs on said complex outputs is utilized to determine the
nature of a successive adjustment of the complex inputs independent
of knowledge of a dynamic structural system characteristic of the
mechanical structure in the form of a matrix of system responses to
known vibrational inputs at the selected harmonics of interest.
13. A method for cancelling vibration according to claim 12,
wherein said step of producing the complex inputs on the basis of
an adaptation cycle is performed utilizing a control algorithm to
determine the complex input, said control algorithm being:
where x.sub.j is a nominal value of the complex input during a
current adaptation cycle j, x.sub.j+1 is the nominal value of the
complex input vector for the next adaptation cycle j+1, E.sup.2 ( )
is the square of the complex output measured with a perturbed
(x.sub.j +u.sub.j) and an unperturbed (x.sub.j) complex input,
u.sub.j is the jth sample of a random vector with covariance
.sigma..sup.2 I, and .beta. is a positive constant used to control
the speed of adjustment of x.
14. A method for cancelling vibration according to claim 13,
wherein the value of x.sub.j+1 is changed in the direction of
u.sub.j when a decrease in E.sup.2 occurs in response to adding of
u.sub.j to the complex input x.sub.j, and the value of x.sub.j+1 is
changed in a direction opposite to u.sub.j when E.sup.2 increases
in response to adding of u.sub.j to the complex input x.sub.j.
Description
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to improvements in methods and apparatuses
for reducing vibration-induced noise in machinery, and in
particular to a method and apparatus for actively cancelling
vibrations in a structure that supports rotating machinery.
2. Prior Art
It is desirable to reduce or eliminate vibrations induced in
rotating machinery. Various active and passive methods have been
employed to suppress vibrations. Examples of passive methods
include cushion supports and mechanical damping means which in
essence absorb and dissipate the vibrational energy produced by the
disturbance. Passive methods are generally unsatisfactory because
the vibrational energy is ultimately transferred to the
environment. This occurs because the vibrational energy contains
complex wave forms which manifest themselves at various harmonics
of the fundamental vibration frequency. A simple mechanical
absorber or damper may thus be essentially transparent to the
various components of the vibrational energy. Accordingly, such
efforts to suppress, cancel or eliminate the vibrations may not be
effective.
Active methods are more successful at eliminating or cancelling
vibrations. However, these too have deficiencies. Like passive
methods, active methods may only be operable within a narrow
frequency range of the fundamental distrubance. In active systems a
counteracting force is produced which opposes the force produced by
the disturbance. The opposing force is not easily generated with
accuracy because the nature of the disturbance is rarely completely
known. The problem is further aggravated by the fact that the
structure may be complex and is not amenable to a simplified rigid
body analysis. Furthermore, most active systems must be custom
designed for a specific structural system and/or type and postion
of forcers in that the specific characteristics of the physical
system as well as those of the forcers must be known. For example,
U.S. Pat. No. 4,626,754 to Habermann et al. discloses a method and
device for reducing the vibrations of rotating machines with an
active suspension which is limited to machines which utilize
electromagnetic bearings that have electromagnets arranged in pairs
along fixed orthogonal diametral axes.
RELATED INVENTIONS
A method for reducing or cancelling vibration induced in rotating
machinery is disclosed in the applicant's copending patent
application Ser. No. 375,227, filed on Jul. 3, 1989. In this
application, the teachings of which are incorporated herein by
reference, unwanted vibration in the mechanical structure 10 caused
by a periodic pulsating force 12 in a rotating shaft 14 can be
cancelled by the arrangement illustrated in FIGS. 1A-C.
A reaction mass actuator or forcer 16, acting on the shaft 14
through a permanent magnet or electromagnet 18 applies a controlled
counteractive force 20 to the shaft 14 which opposes the shaft
pulsation force 12. The actuator 16 operates in response to an
output of adaptive vibration canceller 22. The counteractive force
20 cancels the vibrations in the structure as measured by the
velocity or acceleration sensor 24 which is physically remote from
the forcer 16 as illustrated. The adaptive vibration canceller 22
generates weighted sinusoidal force components which follow the
harmonic frequencies of the shaft pulsation force 12. In the system
described, the rotational speed w of the shaft 14 is measured by an
optical or magnetic incremental encoder 26 which produces output
pulses in synchronism with the rotation of the shaft 14. The output
of the encoder 26 is harmonically related to the shaft pulsation
force 12. Accordingly, shaft rotational speed w and force output 20
are related.
In the arrangement illustrated, a rotational harmonic generator 28
(FIGS. 1A and 1B) responsive to the encoder 26 produces various
time base sinusoidal signals 29 at the fundamental rotational speed
w of the shaft 14 and at harmonics thereof. The time base
sinusoidal signals or outputs 29 of generator 28 are in the form:
e.sup.jkwt, where k is an integer 1, 2, 3 . . . n and w is the
speed of the shaft 14. The outputs of the generators 28 are used to
generate weighted force component signals 30 in adaptive vibration
canceller 22 at the various selected harmonics. The actuator 16 may
thus be controlled by means of adaptive vibration canceller 22,
encoder 26 and the generator 28 at the fundamental shaft rotation
frequency and at various selected harmonics thereof. It is to be
understood that because harmonic frequencies of the force
components are based upon the encoder outputs, the weighted force
components 30 follow the harmonic of the pulsation force or
disturbance 12 as the shaft rotation speed varies.
Other vibration cancellation schemes based on FFT or time-domain
methods would use the same time base, but would not generate the
same sinusoidal waveforms. Any number of harmonics may be employed
to produce the desired force components. For the purpose of this
discussion, only the kth harmonic is illustrated it being
understood that the sum of the various selected harmonics drive the
actuator 16.
In the illustration (FIG. 1C), the disturbance or pulsation force
12 may be represented as a complex number in the form of A sin kwt
and B cos kwt. A and B are unknown coefficients of a single complex
number. The weighted force components 30 are signals which drive
forcer 16 and are also represented in the form C sin and D cos
where C and D are the weighted coefficients of a complex number.
The values of C and D are varied to thereby control the response of
the forcer 16. Sine and cosine components are supplied by the
generator 28 at kth harmonic.
The entire structure 10 has a system dynamic characteristic 32
which is in the form of G<.phi.. where G is the gain at the kth
harmonic represented by the ratio of the accelerometer output 25
over the actuator input 30, and .phi. is the phase angle between
the signals.
In the arrangement of FIG. 1C, the mechanical disturbance 12 is
mechanically combined with the counteractive force 20 of the forcer
16 by interaction with the structure 10. The resulting physical
acceleration E is detected by sensor 24 (e.g., an accelerometer).
The output 25 of sensor 24 is coupled to adaptive vibration
canceller 22 wherein it is multiplied at 36 and integrated over
time at 34 in the preprocessor 40 by the kth harmonic from the
generator 28. The outputs 38 are Fourier coefficients of E in the
form of sin wt and cos wt and feed adaptive algorithm processor 42
which produces weighted components C and D. The components C and D
are combined with the generator outputs to produce weighted force
component outputs 30 for driving forcer 16.
In the arrangement described for one forcer 16 and one sensor 24,
the adaptive algorithm processor 42 solves two linear equations for
the two unknowns A and B which then determine the weighted values C
and D. The combined weighted force component signals C sin kwt, D
cos kwt, 30 drive the forcer 16 at the kth harmonic.
In the current harmonic (or Fourier series) based algorithm the
sine and cosine waveforms at each harmonic frequency are multiplied
by adaptively adjusted weights C and D and are then summed with the
corresponding sines and cosines from the other harmonics to
determine the controlled force 20 applied to the shaft 14 via the
reaction mass actuator or forcer 16. The accelerometer measurement
signal (which must be minimized in an adaptive vibration
cancellation system) is resolved into its Fourier components by
separately multiplying it by the sine or cosine of each harmonic
frequency and integrating the product over an entire cycle of shaft
rotation, obtaining two error signal Fourier coefficients at each
harmonic frequency. The Fourier coefficients at a given frequency
are then used to adjust the actuator adaptive weights at that same
frequency, so as to minimize these error signal Fourier
coefficients themselves.
As long as the mechanical vibrational system is linear, the
adaptation process at one harmonic frequency will not interact with
the adaptation at any other harmonic frequency. These operations
for the kth harmonic summarized in FIG. 1C may thus be combined
with other harmonics of interest.
The arrangement described more fully and in greater detail in the
above-identified application is directed to a single forcer, single
accelerometer system and does not address the problem of reducing
vibrations at various locations in a complex structure. The problem
is complicated by the fact that the number of actuators is usually
fewer than the number of accelerometers. Also, to be most
effective, actuators should be designed into the equipment and not
merely added on. This greatly reduces the number of available
actuator locations. Thus, the cost of such equipment is
considerably increased for each actuator provided.
As a result, in a later co-pending application Ser. No. 551,691,
filed Jul. 2, 1990 of the applicant, a multivariable adaptive
vibration canceller was designed to minimize the total periodically
induced vibratory disturbance in a structure having a unique
dynamic structural characteristic, using a plurality of actuators
and an equal or greater number of accelerometers where each
actuator significantly interacts with each accelerometer to achieve
the lowest possible resulting vibration in accordance with the
structural characteristic. The diclosure of this second application
is also incorporated herein by reference.
In the exemplary embodiment, the actuators are operated in response
to complex input signals to produce counteractive forces in the
structure. Accelerometers sense the combined structural response to
the vibrational disturbance and the counteractive forces, and
produce outputs indicative thereof. An adaptive algorithm processor
which has an electrical characteristic related to the structural
characteristic produces complex outputs for each actuator. The
complex outputs are adjusted by the processor to result in a
structural response to the combined disturbance and the actuator
inputs which minimizes vibration energy at the accelerometer
locations.
In the arrangement of this second application, which is illustrated
in FIG. 2, a system 50 may be a complex structure upon which a
disturbance 52 is imposed. As in the previously described
arrangements, each input may be represented as a composite of sine
and cosine components. For simplicity, however, each input is
represented as a single line. Also, the sinusoidal time base
outputs 29 of harmonic generator 28 (FIG. 1B) are applied as
described in the first related patent application to provide the
harmonic time base of the system.
In the system 50, a plurality of accelerometers or detectors 54 (1,
2, . . . m) are placed at various locations to detect the motion or
acceleration of the structure 50 at such points. A plurality of
forcers or actuators 56 (1, 2, . . . n), where n<m, interact
with the structure 50 to impose counteractive forces 58 thereon as
shown. The sum of the system responses to the disturbance 52 and
the various forcer inputs 64(1) . . . (64(n) is sensed by each of
the accelerometers 54, each of which, in turn, produces a complex
output 60 (1, 2, . . . m). For each harmonic frequency of interest
(in the case illustrated the kth harmonic), a multivariable
adaptive vibration cancellation system or processor 62 produces
weighted complex electrical outputs 64 which drive each of the
forcers 56 such that the output 60 of accelerometers 54 outputs go
to a minimum. Although not shown in detail, the weighted outputs
64'(l) . . . 64'(n) from other processors for other harmonics bf
interest may be combined with the outputs 64(1) . . . 64(n) at the
kth harmonic to drive forcer 56.
The physical system 50 has a dynamic system characteristic which
may be determined by experimental means. For example, each forcer
56 may be activated one at a time by a complex input of a selected
frequency while the physical system 50 is at rest. Outputs 60 of
the various accelerometers 54 may be input to a spectrum analyzer
such as a model 1172 Frequency Response Analyzer manufactured by
Schlumberger, Inc. Each forcer 56, thereby produces a resulting
vibration in the physical system which produces a corresponding
measurable output at each accelerometer.
Each forcer 56 may be operated separately at the particular input
frequency of interest and each accelerometer output 60 may be
separately analyzed to compile a matrix of data for various
frequencies and harmonics of such frequency. The various forcer
inputs are selected in anticipation of the rotational speeds at
which it is expected that the equipment will be operated. In the
preferred embodiment, the equipment is not operated while the
measurements are made. For each forcer input at a selected
frequency of interest, the physical system can be represented by an
m by n transfer function matrix [A] of complex numbers. When 2
forcers and 4 accelerometers are used a 4 by 2 matrix results.
The mechanical structure 50, as represented by the 4 by 2 matrix
[A] of complex numbers of the kth harmonic relates the Fourier
input coefficients of the forcers 56 (1, 2, . . . n) to the Fourier
output coefficients of the accelerometers 54 (1, 2, . . . m).
During operation, each accelerometer 54 measures the vibrational
disturbance 52 at its location as well as the effect of each forcer
56 at such location, so that the m, kth harmonic accelerometer
Fourier output coefficients 60 (1, 2, . . . m) may be combined as a
multivariable adaptive vibration cancellation complex vector E. The
values of E are processed in the adaptive vibration cancellation
processor 62 to produce the forcer inputs 64 by means of a matrix
algebra algorithm. E represents the error signal (algebraic sum) or
difference between the disturbance and all the counteractive forces
on the structure 50.
In accordance with this earlier invention, the components of the
complex numbers of the transfer function matrix [A] are known for
all harmonic frequencies at which vibration cancellation is to
occur. Because shaft rotation speed can change, a frequency
response matrix is generated over a wide range of frequencies of
interest for the mechanical structure with respect to the forcer
and accelerometer locations as noted above. Further, because the
mechanical structure is complicated, there is a significant
interaction between every forcer and every accelerometer.
Accordingly, the adaptive vibration canceller 62 must use all four
accelerometer output signals 60 to obtain the two forcer signals 64
in a single coordinated multi-input, multi-output algorithm.
The kth harmonic relation between the forcers 56, the
accelerometers 54, the disturbance 52 and the physical system 50
may be summarized by the following expression:
Where E is the vector of four complex numbers which are the kth
harmonic Fourier coefficients of the four accelerometers 54; b is
the vector of the coefficients representing the vibrational
disturbance 52 acting on the four accelerometers 54 which is an
unknown vector input; [A] is the complex 4 by 2 physical system
transfer function matrix; and x is the vector of two forcer Fourier
coefficients. The terms on the right hand side of the expression,
namely, b and Ax, express the two sources of vibration measured by
the accelerometers, namely, the vibrational disturbance b which is
unknown and the forcer motion Ax which is established by the
algorithm processor output. The expression E is the total vibration
energy at the accelerometer locations. It is intended that the
magnitude of the accelerometer vector E should be minimized by
employing a sum of the squares operation on the accelerometer
coefficients.
A variety of mathematical processes may be employed to achieve the
desired minimization of the vector. However, in the preferred
embodiment the vector of forcer signals 64 which act to minimize
the magnitude of the accelerometer vector E are resolved by solving
the so called normal equations for x using the equation:
where [A.sup.*T ] is a 2 by 4 matrix, known as a complex conjugate
transpose of [A].
At the frequency of the kth harmonic, the matrix [A] is empirically
determined. Matrix [A.sup.*t ] is determined by the mathematical
transposition. Once the harmonic is known, the forcer vector x can
be determined, because the output of the adaptive vibration
cancellation processor 62 is thus specified by mathematical
operation. Although the vector b representing the disturbance is
not known, the vector E=b-Ax is measured directly by the
accelerometers.
In FIG. 2, the adaptive vibration cancellation processor 62
includes processing matrix 68 which effects the complex conjugate
transpose of A. The A.sup.*t values or instantaneous forcer inputs
are separately processed in respective integrators 70 (1, 2, . . .
n) in order to produce values 64 (1, 2, . . . n) which are
solutions for x of equation [2] obtained through complex vector
matrix integral equations described in the second application. The
harmonic generator time base sinusoidal outputs 29 combine with
integrators 70 to produce the complex solution. The discrete time
equivalent of integration must be used for the integrators since
the accelerometer Fourier coefficients are updated at the end of
each cycle of shaft rotation.
In order to minimize vibration, the adaptive vibration cancellation
processor 62 must produce values of x which converge to a solution
of the normal equation [2]. From the foregoing, it can be
recognized that knowledge of the transfer function [A] of the
physical system must be available to utilize the adaptive vibration
canceller of applicant's second application. On the other hand,
circumstances exist where such information is not available or is
not easiy obtained.
SUMMARY OF THE INVENTION
In view of the foregoing, it is the primary object of the present
invention to provide an improved adaptive vibration canceller
system by which it is possible to minimize vibration in a
mechanical structure for which there are many forcers and
accelerometers, and the transfer function of the physical system is
unknown.
Another object of the invention is to provide an improved vibration
canceller system which is applicable to a wide range of specific
applications utilizing various types and numbers of forcers and
sensors;
It is a further object to provide an improvement in adaptive
vibration cancelling systems for physical systems in the form of
mechanical structures having a rotary shaft.
In accordance with the present invention, this object is achieved
by utilizing an adaption period to determine the effect of
variation of the forcer vector on the error signal output from the
accelerometers. The observed change in the size and direction of
change in the error signal is then utilized to determine the
nominal value of the forcer vector for use in a vector difference
equation of the present invention, by which the adjustment of the
forcers for the next cycle is performed. By repeating such
operations, the adjustments produce a convergence of the effects
toward the minimum total vibration for the mechanical structure
without any knowledge of its transfer function.
The further objects, details and advantages of the present
invention will become more apparent from the following detailed
description when viewed in conjunction with the accompanying
drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1A is a system block diagram of a first related application of
the applicant;
FIG. 1B is an electrical block diagram of a harmonic generator of
the FIG. 1A system;
FIG. 1C is an electrical block diagram of the basic adaptive
vibration canceller of FIG. 1A;
FIG. 2 is an electrical block diagram of a multivariable adaptive
vibration canceller of a second related application of the
applicant; and
FIG. 3 is an electrical block diagram of an improved adaptive
vibration canceller in accordance with the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
As noted above, the kth harmonic relation between the forcers, the
accelerometers, the disturbance, and the physical system shown in
FIG. 2 can be summarized by equation [1] where E is (for the system
shown in FIG. 2) the vector of four complex numbers which are the
kth harmonic Fourier coefficients of the four accelerometers, b is
the vector of four coefficients showing the vibrational disturbance
52 acting on the four accelerometers 54 (the value of this vector
is unknown), A is the complex 4.times.2 physical system transfer
function matrix, and x is the vector of two forcer Fourier
coefficients. The two terms on the right-hand side of equation [1],
b and Ax, express the two sources of vibration measured by the
accelerometers: b, the vibrational disturbance, and Ax, the forcer
motion. In the FIG. 2 system, the magnitude of the accelerometer
vector E is minimized by performing a sum of the squares operation
on the accelerometer coefficients using a mathematical process
requiring knowledge of the transfer function [A].
However, in the present case, instead of requiring such knowledge,
use is made of the more basic fact that minimizing the magnitude of
the accelerometer vector E is the same as minimizing the sum of the
squares of the accelerometer coefficients, and is, thus, the same
as minimizing the total vibration energy at the accelerometer
locations. That is the sum of the squares of the accelerometer
coefficients, which is also the squared-magnitude of the
accelerometer error vector, can from equation [1] be expressed
as:
where H denotes the Hermitian (complex conjugate transpose) of b,
A, x, or E. A.sup.H is (in a two forcer, four accelerometer
example) a 2.times.4 matrix.
For the improved vibration canceller, the matrix A does not have to
be known. Once it is decided upon, the forcer vector x can be
known, because it is the output of the vibration canceller control
processor 71, and is, thus, specified by its control algorithm 72.
Although the vector of disturbance b is not known, the error vector
E=b-Ax is measured directly by the accelerometers and can be used
to form the squared-error E.sup.2 (x), which is to be
minimized.
As can be seen from equation [3], E.sup.2 (x)is a quadratic
function of the forcer vector x and so can be visualized as a
paraboloidal surface in a higher-dimensional space of forcer
coefficients. Since it is a quadratic function of x, the surface
E.sup.2 (x) has only one local minimum with respect to x, and this
local minimum is also the global minimum. This means that an
algorithm that incrementally adjusts x to produce incremental
decreases in E.sup.2, will not get trapped on a local minimum that
is different than the global minimum.
Since the system's transfer function matrix [A] is not known,
measurement of E.sup.2 (x) at a single operating point is not
enough information with which to determine which way the forcer
vector x should be adjusted to decrease E.sup.2 (x). To decide the
correct direction of adjustment, an adaptation period which
consists of two shaft rotation cycles is used. During the first
shaft rotation, the forcer vector is held at the current nominal
value x, and the accelerometers' squared-error E.sup.2 (X) is
measured. During the second shaft rotation, the forcer vector x is
perturbed by a zero-mean random vector u with a diagonal covariance
of .sigma..sup.2 I and the squared-error E.sup.2 (x+u) is measured,
where I is the unit or identity matrix. The size and the direction
of the change in E.sup.2 observed during the current adaptation
period will determine the nominal value of x for the next
adaptation period, according to the following vector difference
equation which is the subject of the present disclosure:
where x.sub.j is the nominal value of the forcer vector during the
current adaptation period j, x.sub.j+1 is the nominal value of the
forcer vector for the next adaptation period j+1, E.sup.2 () is the
square of the accelerometer vector measured with either the
perturbed (x.sub.j +u.sub.j) or unperturbed (x.sub.j) forcer
vector, u.sub.j is the jth sample of a random vector (i.e., u.sub.j
is composed of a pair of complex numbers in a two-forcer example)
with covariance .sigma..sup.2 I, where I is the unit or identity
matrix, and .beta. is a positive constant used to control the speed
of adjustment of x. The use of this algorithm to minimize .the
vibration energy at the accelerometers is shown in FIG. 3.
From equation 4, if E.sup.2 decreases when u.sub.j is added to the
forcer vector, then the nominal value of the forcer vector x is
changed in the direction of u.sub.j. If E.sup.2 increases when
u.sub.j is added, then x is changed in the direction opposite to
u.sub.j. The change in the nominal value of x will, in either case,
continue to reduce the acceleration energy E.sup.2 until the unique
local and global minimum-acceleration condition is reached, all
without any knowledge or use of the mechanical structure's transfer
function matrix [A].
This means that the improved vibration canceller shown in FIG. 3,
with the processing control algorithm given in equation [4], will
minimize the vibration energy measured at the accelerometers.
For completeness, it should be recognized that the value of E.sup.2
in equation [4] is a scalar despite the complex nature of the
outputs from the accelerometer sensors 54 and despite the complex
nature of the inputs to the forcers 56. That is, for each
accelerometer output a sine and cosine multiplication and
integration is performed in the manner described above relative to
FIG. 1C at 36, 34, respectively. In processor 71, each of these
output values is multiplied by its own complex conjugate, and then
all of these values are summed with this sum used as the E.sup.2 in
the control algorithm. Also, with regard to the outputs x to each
forcer 56, while shown in FIG. 3 as a single line, as reflected by
the sine and cosine labels, each output, in the manner described
relative to applicant's first related application, is multiplied by
sine and cosine weighting components produced by the harmonic
generator 28 before being utilized to drive the respective forcer
56.
From the foregoing, it should now be apparent how the disclosed
improved vibration canceller of the present invention is capable of
minimizing vibration in a mechanical structure for which there are
(a) many forcers and many accelerometers and (b) no knowledge of
the mechanical structure's transfer function.
It should be appreciated that the above described preferred
embodiment of the invention is merely exemplary in nature, and
various modifications and changes will be apparent to those of
ordinary skill in the art. Thus, the invention should not be
considered as being limited to those aspects specifically mentioned
and, instead, includes all modifications, alterations, variations,
and changes as are encompassed by the scope of the appended
claims.
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